How to place objects on top of another object and join their geometry
I've been doing this quite a bit lately, but BEWARE, I'm very much a newbie myself, so hopefully you'll get better answers from others.
To join that dome to the cylinder here's what I would do.

I would fist make a duplicate of both of the two pieces (using shift-D) on each, and then I'd hide the original pieces. This is so that I can always just delete the copies, and start over with the original pieces if something gets fouled up. Don't bother renaming the copies yet.
One mistake you really don't want to make is to join two pieces like the one in your example that will have faces touching once you join them! Look at the bottom of the dome. That face on the bottom of the dome, once you join the two pieces, will occupy precisely the same 3 dimensional space as the top face of the cylinder. That will confuse the dickens out of blender. So...
With the copy of the dome, delete the faces that comprise the bottom surface of it. (Delete them, don't dissolve them). Then do the same thing with the faces on the top surface of the cylinder copy.
In the window on the lower right in your example you appear to be in the modifier tab. Switch to the scene tab (the one that looks like a cone with a tiny sphere next to it). If you working in metric, switch to millimeters, if you're working in feet, switch to inches. This will give you finer dimensions to work with.
(since you seem to be having trouble with snapping I'll try to minimize using it, but you'll have to learn how to use snapping sooner or later. Watch a bunch of YouTube videos on snapping, trust me snapping is simply too useful to continue to try to do without. That said I'll try to use snapping as little as possible.)
Engage the measure tool. That's the bottom icon in you're main window that looks like the letter L with tiny lines on it. Go into wire-frame mode, and make sure the x-ray function is off. Click and hold near where you want to begin the measurement, drag to near where you want to end the measurement (kind of like pulling out a tape measure).
Now here's where you'll have to use snapping.
In you're example you appear to be set to snap to faces, switch to snapping to vertex.
Look for a corner on the bottom of the dome that is directly above a corner on the cylinder. (any corner will have a vertex to snap to)
Click and hold on one end of the measuring tool you made. Drag it a short distance in any direction, and while still holding the mouse button down hold the control button on your keyboard down (this engages the snapping action). Still holding the mouse button and control key down, move the end of the measuring tool to the corner where you want the measurement to start and when it gets close to that corner's vertex it will snap to that vertex, and you can let go. You now have one end of the measurement tool anchored to the fist corner. Use exactly the same procedure to snap the other end of the measure tool to a corner directly below the first corner. Check using the front view, and then a side view to make certain that the measurement tool is vertical and not at some angle that will throw off you measurement.
Now you have a precise measurement of how far you have to move either the cylinder, or the dome in the Z direction to get them to meet. Write it down!
I assume you know how to move them once you know how far to move them.
Once you've completed the move you can now join the two.
Once you've joined them, go into edit mode, select all (I use the hotkey A to do that), then go into Mesh, Clean-up, Merge by Distance and set the distance to more than the default, but much less than the shortest distance between any vertices. (your diameter is 42mm, so set the distance in the merge to say 5mm. If lines start jumping all over the place it means your merge distance is too high. Keep decreasing the merge distance until things merging that you don't want to merge stop doing it.
Keep an eye on the little blue boxed note the pops up at the bottom of the screen, it will tell you how many vertices where merged. If you keep seeing 0 vertices merged, that's not good. Think about it. The bottom of the dome and the top of the cylinder form circles that are essentially identical. Which means when they are joined you're now going to have 2 vertices at each corner making up those circles.  That's going to give you bad geometry. So keep increasing the merge distance until the little blue pop-up window shows that you've merged at least a dozen or so vertices.

Decrease the merge distance whenever it deforms your geometry, Increase the merge distance whenever you get a value of 0 vertices merged.
Many times, like when you use booleans, you will sometimes get 0 vertices merged, and it's ok, because there weren't any vertices on top of each other that need to be merged. But it this particular case you know for a fact that there are going to be vertices on top of each other because you just finished joining the circles at the bottom of the dome to the circles making up the top of the cylinder. So there has to be some overlapping geometry there.
You can keep changing the merge distance until you click on something. If you do a merge that fouls things up, there's always Control-Z to undo it.
Once you've finished, and the two copies are now one piece, it's finally a good time to rename your new object.
Hope this helps, Mike.